Group #34101118 — subgroups

Subgroups — #34101118

Order 20096.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757297	C₂	1	1	yes	maximal, characteristic
2	#23757119	C₂	1	1	yes	maximal, characteristic
3	#23757032	C₂	1	4	yes	maximal
4	#533×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
7	C₄⋊#11929434	—	2	2	no	—
8	#19097678	—	2	2	no	—
9	C₄⋊Q₈×C₁₅₇	C₄	1	2	yes	complemented
10	#125×C₁₅₇	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₁₅₇	—	2	2	no	—
12	C₄×Q₈×C₁₅₇	—	2	2	no	—
13	#117×C₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	—	2	2	no	—
16	#14578938	—	2	4	no	—
17	C₂×#11929434	—	2	2	no	—
18	C₄×Dic₁₅₇	—	2	2	no	—
19	C₄×Dic₁₅₇	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₁₅₇	—	2	2	no	—
21	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₅₇	—	2	4	no	—
24	C₄⋊C₄×C₁₅₇	—	2	2	no	—
25	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
26	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄²×C₁₅₇	—	2	2	no	—
28	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₅₇	D₄	1	2	yes	—
31	C₂×Dic₁₅₇	—	2	4	no	—
32	C₂×Dic₁₅₇	—	4	4	no	—
33	#11929434	—	4	2	no	—
34	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	Q₈×C₁₅₇	—	4	4	no	—
36	C₂×C₄×C₁₅₇	—	2	4	no	—
37	C₂×C₄×C₁₅₇	—	4	2	no	—
38	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₅₇	#41	1	2	yes	—
44	C₂²×C₁₅₇	—	2	2	no	—
45	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₅₇	—	2	2	no	—
48	C₄×C₁₅₇	—	2	2	no	—
49	C₄×C₁₅₇	—	4	4	no	—
50	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
51	C₄×C₁₅₇	#124	1	1	yes	characteristic
52	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₅₇	—	4	4	no	—
54	C₂×C₁₅₇	—	2	2	no	—
55	C₂×C₁₅₇	#391	1	1	yes	characteristic
56	C₂×C₁₅₇	#421	1	2	yes	—
57	C₁₅₇	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	157	1	no	maximal, Sylow
59	#533	D₁₅₇	1	1	yes	characteristic
60	#423	—	157	1	no	—
61	#376	—	157	1	no	—
62	#339	—	157	4	no	—
63	C₂×M₁₆	—	157	2	no	—
64	C₄⋊Q₈	Dic₁₅₇	1	2	yes	complemented
65	#125	Dic₁₅₇	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₁₅₇	1	2	yes	—
69	C₂×C₄²	—	157	2	no	—
70	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
71	C₄⋊C₈	—	314	2	no	—
72	#95	—	314	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	314	2	no	—
75	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
77	M₁₆	—	314	4	no	—
78	C₂×C₈	—	314	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	314	2	no	—
85	C₄²	—	157	4	no	—
86	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	314	4	no	—
94	C₂×C₄	—	628	4	no	—
95	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
97	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
100	C₂×C₄	#14578955	1	2	yes	—
101	C₈	—	628	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
104	C₂²	#19097747	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	628	4	no	—
109	C₄	Q₈×Dic₁₅₇	1	2	yes	—
110	C₄	#19097760	1	1	yes	characteristic
111	C₄	#19097769	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#23757180	1	1	yes	characteristic
114	C₂	#23757290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757297

C₂

yes

maximal, characteristic

#23757119

C₂

yes

maximal, characteristic

#23757032

C₂

yes

maximal

#533×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₄⋊#11929434

—

#19097678

—

C₄⋊Q₈×C₁₅₇

C₄

yes

complemented

#125×C₁₅₇

C₄

yes

complemented

C₂²⋊Q₈×C₁₅₇

—

C₄×Q₈×C₁₅₇

—

#117×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

#14578938

—

C₂×#11929434

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

Q₈

yes

complemented

C₂×Q₈×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#391

yes

characteristic

C₂×C₁₅₇

#421

yes

—

C₁₅₇

#1995

yes

characteristic, complemented, Sylow

#1995

—

157

maximal, Sylow

#533

D₁₅₇

yes

characteristic

#423

—

157

—

#376

—

157

—

#339

—

157

—

C₂×M₁₆

—

157

—

C₄⋊Q₈

Dic₁₅₇

yes

complemented

#125

Dic₁₅₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

314

—

#95

—

314

—

C₂×Q₈

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

Q₈×D₁₅₇

yes

—

100

C₂×C₄

#14578955

yes

—

101

C₈

—

628

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

104

C₂²

#19097747

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

628

—

109

C₄

Q₈×Dic₁₅₇

yes

—

110

C₄

#19097760

yes

characteristic

111

C₄

#19097769

yes

characteristic

112

C₂

—

113

C₂

#23757180

yes

characteristic

114

C₂

#23757290

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#533×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#533	D₁₅₇
hypocenter	C₁₅₇	#1995
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#533×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₁₅₇	C₄	semidirect
10. #125×C₁₅₇	C₄	semidirect
19. C₄×Dic₁₅₇	Q₈	semidirect
57. C₁₅₇	#1995	semidirect
64. C₄⋊Q₈	Dic₁₅₇	semidirect
65. #125	Dic₁₅₇	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₁₅₇

C₄

semidirect

10. #125×C₁₅₇

C₄

semidirect

19. C₄×Dic₁₅₇

Q₈

semidirect

57. C₁₅₇

#1995

semidirect

64. C₄⋊Q₈

Dic₁₅₇

semidirect

65. #125

Dic₁₅₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23757297	2
#23757119	2
#23757032	2
#533×C₁₅₇	2
#1995	157

subgroup

index

#533×C₁₅₇

157

Subgroups — #34101118

Conjugacy classes of subgroups

Complements

Maximal subgroups